1549301742-The_Theory_of_Difference_Schemes__Samarskii

240 Difference Schernes for Elliptic Equations

if v(x) is an arbitrary function having no less than four derivatives with

respect to xa, Cl'= 1, 2, that are bounded at least in the rectangle {xa-ha <

x~ < x a + ha; Cl' = 1, 2} for ha < ha. So, the Laplace operator (2) is

approximated to second order by the difference operator (5) on a regular

"cross" pattern. A difference approximation of the p-dimensional (p > 2)

Laplace operator

(9)

p

Lu= L Lau,

a=l

82 u

Lau=~,

uxa

can be arranged in just the same way. This can be done by replacing La

by the three-point difference operator Aa and accepting the decomposition

(10)

so that

(11)

p

Av= L Aa v,

a=l

Aa V = Vx-0: ,{,a-~ ,

where v(±l"') = v(x(±l"J). Here x(+l.,) (or x(-l"')) is a point into which

the point x = (x 1 , ... , xp) moves after the shift by one interval ha along

the direction xa to the right (or to the left) (see Fig. 7).

x

Figure 7.

Evidently, the pattern for operator (10) consists of 2p + 1 points: x,

x(±la), Cl'= 1, ... ,p (7 points in the case p = 3) and the approximation

here is of order 2.

2. Approximation of the Laplace operator on an irregular "cross" pattern.
   We now consider a difference approximation of the Laplace operator on
   an inegular "cross" pattern. In the two-dimensional case (p = 2) such a
   pattern consists of the five points